Stuck with finding roots including population models

Hi,

I would like to find the solution for t in;

a*t=1/(1+c*exp(-b*t))

where a, b and c are all positive and less than 1. Left hand side of the function is cumulative logistics curve. I know it is not invertible, I am assuming the solution would include LambertW function. Is there a way to find t in closed form (using LambertW or any other functions)?
Thanks,
rutabaga

Quote:

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Originally Posted by rutabaga

. . . or any other functions)?
Thanks,
rutabaga

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Here's a numerical approach in Mathematica:

Code:

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f[a_, b_, c_, t_] := a^t - 1/(1 + c/E^(b*t));
g[x_, y_, z_] := t /. FindRoot[f[x, y, z, t] == 0, {t, 0}]

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